On Revisiting Gödel's Incompleteness and Turing's Undecidability of the Halting Problem

This paper establishes a rigorous formal foundation for the interplay betweenrecursive function theory, Turing computability, and Peano Arithmetic. We beginby constructing the class of recursive functions via closure operations, proceedingto formalize their arithmetization through Gödel numbering and the EnumerationTheorem. Building upon the Church–Turing Thesis, we demonstrate the existenceof a universal partial recursive function and its corresponding Universal Turing Machine. The paper then transitions into the formal system of Peano Arithmetic,where we develop a detailed Gödel numbering of the language and metamathematical predicates, enabling the internal representation of syntax and proof structures.Key results include the formalization of Kleene’s Normal Form Theorem within PAand a comprehensive analysis of the representability of recursive functions and predicates. In later sections, we undertake a granular re-examination of self-referencemechanisms, exploring the structural behavior of diagonalization under varying logical constraints. This work serves as the first in a series of investigations into thelogical, and philosophical foundations of mathematics, computability and formalsystems.